Space-filling lattice designs for computer experiments

TL;DR

This paper develops algorithms for constructing space-filling quasi-Monte Carlo lattice designs, demonstrating their theoretical properties and practical effectiveness for computer experiments.

Contribution

It introduces two novel algorithms for generating quasi-uniform lattice designs and validates their theoretical and empirical performance.

Findings

Explicit construction of isotropic discrepancy $O(N^{-1/d})$

Validation of quasi-uniformity through numerical experiments

Demonstrated effectiveness in Gaussian process regression

Abstract

This paper investigates the construction of space-filling designs for computer experiments. The space-filling property is characterized by the covering and separation radii of a design, which are integrated through the unified criterion of quasi-uniformity. We focus on a special class of designs, known as quasi-Monte Carlo (QMC) lattice point sets, and propose two construction algorithms. The first algorithm generates rank-1 lattice point sets as an approximation of quasi-uniform Kronecker sequences, where the generating vector is determined explicitly. As a byproduct of our analysis, we prove that this explicit point set achieves an isotropic discrepancy of $O(N^{-1/d})$. The second algorithm utilizes Korobov lattice point sets, employing the Lenstra--Lenstra--Lov\'{a}sz (LLL) basis reduction algorithm to identify the generating vector that ensures quasi-uniformity. Numerical experiments are provided to validate our theoretical claims regarding quasi-uniformity. Furthermore, we conduct empirical comparisons between various QMC point sets in the context of Gaussian process regression, showcasing the efficacy of the proposed designs for computer experiments.